Does the reality of evil mean that God does not exist? The answer is no. That's because evil actually provides a good reason to believe God exists. In fact, if God does not exist then evil cannot exist; if evil is real then God must be real. How is this possible? Consider the following.
Suffering and pain can exist apart from God, but these are not the same thing as "evil." When someone finishes running a marathon, they may be in great pain, but that pain is not considered "evil." What this shows is that some pain and suffering is not "evil."
When we call certain suffering "evil," it implies a judgment about the suffering. This judgment amounts to the conviction that things are "not the way they are supposed to be." This, in turn, entails that there is a way that things are supposed to be. Furthermore, the way people make the judgment shows that they do not believe it is based in anyone's opinion. As such, it must be based in a reality outside of any human minds. So what other options are there? The physical world of air, rocks, trees, stars, etc. is an option. But since physical objects do not determine how anything should or should not be, the judgment cannot be based in the physical world. Therefore the "way things are supposed to be" must be based in a reality outside of the physical world. And it is this other reality which can be called "God." (This very brief explanation is developed further in the short video below.)
Dr. Richard Dawkins, a well-known atheist, indirectly acknowledges this point when he states that in an atheistic universe filled with "blind physical forces and genetic replication... there is, at bottom... no evil and no good".
In a universe of blind physical forces and genetic replication, some people are going to get hurt, other people are going to get lucky, and you won't find any rhyme or reason in it, nor any justice… there is, at bottom, no design, no purpose, no evil and no good, nothing but blind, pitiless indifference.
Hence there is a certain irony behind those who reject God because of evil. The irony is that far from disproving the existence of God, our recognition of evil's existence provides a strong basis to conclude that God exists. Of course, why God allows evil is a different challenge. There are various possibilities which are explored in the latter half of the video found here. But regardless of whether someone finds a satisfactory answer to this other question, it has little bearing on the fact of God's existence.
Short Blog: Alvin Plantinga Solves "Logical Problem" of Evil
Video (below): Evil Requires Trusting God's Goodness
Richard Dawkins, River out of Eden: A Darwinian View of Life, Reissued, A Phoenix Paperback (London: Phoenix, 2004), 133.